Generally speaking, resources are invested, and only spent/lost in unusual situations. The other big gotcha is that the system is abstracted out - there are no maintenance costs, and you don't need to track income and payments, except at the highest abstract level.
When a house is created, you get the dice rolls, plus various modifiers for realm and house historical events, to yield the starting resources for the house.
From then on, there are four things that can change them - 1. House Fortunes, 2. House Actions, 3. Narrator events, 4. Player Actions
1. House Fortunes
These represent the general luck/progress/happenings for the house in a given month. At the start, the Head of House (or his acting Steward) rolls a Status-Stewardship (plus modifiers for various House Holdings) and compares the result to the House Fortunes chart. Using the chart as a guide, the Narrator picks one or two resources to be improved or reduced by the indicated amount. To my knowledge, there's really no real guide on which resources should be modified - it's strictly Narrator fiat, but most people seem to agree that some sort of story element should be involved, even if only peripherally.
- Ex: Players roll and get Blessing, indicating either one +1d3 or two +1s. The Narrator decides that the House's mines have been particularly fruitful this month, and the House earns +1d3 to its Wealth.
- Ex: The next month, the players roll and get Disaster, with its one -1d6 or two -1d3s. Deciding to foreshadow an Ironborn invasion, and decides a slew of powerful storms batter the House's coastal domains, destroying infrastructure and causing the people to flee, applying the -1d6 to Population.
2. House Actions
These represent the specific intentions of the players in the roles as leaders of the House. Some of these actions have direct (or indirect) effects on the house resources - such as the manage resources action, which lets the players directly reassign resources of one type to another. Assigning a story element to these actions is typically the duty of both players and Narrator.
- Ex: The players, wanting to help repopulate the flooded areas, send in some of their army to clean things up, using the Manage Resources action and spending some of their uninvested Power to increase their Population.
- Ex: The next month, one of the PCs, having found religion, has convinced the rest that they should build a sept in their small hall, and the house takes the Begin Project action. They invest 15 of their unspent Wealth and then in 12+2d6 months, will finally begin reaping the benefits of a sept. They don't need to spend anything else to accomplish this - consider the 15 Wealth as having being put aside in a "Project fund", and all day-to-day costs associated with building the sept come from this, including stuff like salaries for the project manager, foreman, masons, carpenters, laborers, materials, etc - and they also don't need to take any more actions towards seeing it completed. They start it one month, and then never need think of it again until the 12+2d6 months is up, and then they have a Sept.
3. Narrator Events
Although somewhat similar to the House Fortunes, these are essentially the results of direct NPC actions towards the house, either beneficial or malicious. These are the result of rival houses attacking to seize land, or the dowry and prestige that comes with a daughter of a major house marrying the players' houses heir. Except for the conquest stuff, there are no hard-and-fast rules for how these might modify resources (the Warfare chapter indicates how resources are increased and decreased due to the gain or loss of domains). A lot of these can be rolled into the House Fortunes (ie, if they rolled a Blessing already, then the Narrator can just say that the +1 Influence is from the prestige and the +1 Wealth is the dowry), but some times you might want them separate, so... there you are.
- Ex: The foreshadowed Ironborn raids come, and the longships strike quickly, stealing a coastal domain. If they stay, the players' house loses the land domain - and the invested Lands resource - and some other resources are modified as well. At the same time, the Ironborn house's resources are modified as well, usually with increases. Note that retaliation will require the players' house to use the Wage Wars house action in a later month.
4. Player Actions
These represent those specific actions individual characters take to improve their house's lot in Westeros. Typically, this is through the investment of their rewards back into the house (IE: coin or glory).
- Ex: The players have looted an Ironborn longship returning from reaving elsewhere, and have found its hull filled with casks of Arbor gold and Dornish reds. The Narrator has decided that the wines are altogether worth 200 gold dragons, and for their daring, has awarded each player 1 point of Glory. The players, figuring their characters aren't complete drunks, decide to "store" the wines at their keep - investing the 200gd in their house for +1 Wealth, and some of them decide to invest their Glory in the house as well - in this case, into Influence and Power to represent the men-at-arms who flock to their banner to serve under such brave and skilled warriors.